553 research outputs found
Transverse instability for non-normal parameters
We consider the behaviour of attractors near invariant subspaces on varying a
parameter that does not preserve the dynamics in the invariant subspace but is
otherwise generic, in a smooth dynamical system. We refer to such a parameter
as ``non-normal''. If there is chaos in the invariant subspace that is not
structurally stable, this has the effect of ``blurring out'' blowout
bifurcations over a range of parameter values that we show can have positive
measure in parameter space.
Associated with such blowout bifurcations are bifurcations to attractors
displaying a new type of intermittency that is phenomenologically similar to
on-off intermittency, but where the intersection of the attractor by the
invariant subspace is larger than a minimal attractor. The presence of distinct
repelling and attracting invariant sets leads us to refer to this as ``in-out''
intermittency. Such behaviour cannot appear in systems where the transverse
dynamics is a skew product over the system on the invariant subspace.
We characterise in-out intermittency in terms of its structure in phase space
and in terms of invariants of the dynamics obtained from a Markov model of the
attractor. This model predicts a scaling of the length of laminar phases that
is similar to that for on-off intermittency but which has some differences.Comment: 15 figures, submitted to Nonlinearity, the full paper available at
http://www.maths.qmw.ac.uk/~eo
Affective iconic words benefit from additional soundâmeaning integration in the left amygdala
Recent studies have shown that a similarity between sound and meaning of a word (i.e., iconicity) can help more readily access the meaning of that word, but the neural mechanisms underlying this beneficial role of iconicity in semantic processing remain largely unknown. In an fMRI study, we focused on the affective domain and examined whether affective iconic words (e.g., high arousal in both sound and meaning) activate additional brain regions that integrate emotional information from different domains (i.e., sound and meaning). In line with our hypothesis, affective iconic words, compared to their nonâiconic counterparts, elicited additional BOLD responses in the left amygdala known for its role in multimodal representation of emotions. Functional connectivity analyses revealed that the observed amygdalar activity was modulated by an interaction of iconic condition and activations in two hubs representative for processing sound (left superior temporal gyrus) and meaning (left inferior frontal gyrus) of words. These results provide a neural explanation for the facilitative role of iconicity in language processing and indicate that language users are sensitive to the interaction between sound and meaning aspect of words, suggesting the existence of iconicity as a general property of human language
On the rate of quantum ergodicity in Euclidean billiards
For a large class of quantized ergodic flows the quantum ergodicity theorem
due to Shnirelman, Zelditch, Colin de Verdi\`ere and others states that almost
all eigenfunctions become equidistributed in the semiclassical limit. In this
work we first give a short introduction to the formulation of the quantum
ergodicity theorem for general observables in terms of pseudodifferential
operators and show that it is equivalent to the semiclassical eigenfunction
hypothesis for the Wigner function in the case of ergodic systems. Of great
importance is the rate by which the quantum mechanical expectation values of an
observable tend to their mean value. This is studied numerically for three
Euclidean billiards (stadium, cosine and cardioid billiard) using up to 6000
eigenfunctions. We find that in configuration space the rate of quantum
ergodicity is strongly influenced by localized eigenfunctions like bouncing
ball modes or scarred eigenfunctions. We give a detailed discussion and
explanation of these effects using a simple but powerful model. For the rate of
quantum ergodicity in momentum space we observe a slower decay. We also study
the suitably normalized fluctuations of the expectation values around their
mean, and find good agreement with a Gaussian distribution.Comment: 40 pages, LaTeX2e. This version does not contain any figures. A
version with all figures can be obtained from
http://www.physik.uni-ulm.de/theo/qc/ (File:
http://www.physik.uni-ulm.de/theo/qc/ulm-tp/tp97-8.ps.gz) In case of any
problems contact Arnd B\"acker (e-mail: [email protected]) or Roman
Schubert (e-mail: [email protected]
Classical and quantum ergodicity on orbifolds
We extend to orbifolds classical results on quantum ergodicity due to
Shnirelman, Colin de Verdi\`ere and Zelditch, proving that, for any positive,
first-order self-adjoint elliptic pseudodifferential operator P on a compact
orbifold X with positive principal symbol p, ergodicity of the Hamiltonian flow
of p implies quantum ergodicity for the operator P. We also prove ergodicity of
the geodesic flow on a compact Riemannian orbifold of negative sectional
curvature.Comment: 14 page
Exploring the measurement of markedness and its relationship with other linguistic variables
Antonym pair members can be differentiated by each word's markedness-that distinction attributable to the presence or absence of features at morphological or semantic levels. Morphologically marked words incorporate their unmarked counterpart with additional morphs (e.g., "unlucky" vs. "lucky"); properties used to determine semantically marked words (e.g., "short" vs. "long") are less clearly defined. Despite extensive theoretical scrutiny, the lexical properties of markedness have received scant empirical study. The current paper employs an antonym sequencing approach to measure markedness: establishing markedness probabilities for individual words and evaluating their relationship with other lexical properties (e.g., length, frequency, valence). Regression analyses reveal that markedness probability is, as predicted, related to affixation and also strongly related to valence. Our results support the suggestion that antonym sequence is reflected in discourse, and further analysis demonstrates that markedness probabilities, derived from the antonym sequencing task, reflect the ordering of antonyms within natural language. In line with the Pollyanna Hypothesis, we argue that markedness is closely related to valence; language users demonstrate a tendency to present words evaluated positively ahead of those evaluated negatively if given the choice. Future research should consider the relationship of markedness and valence, and the influence of contextual information in determining which member of an antonym pair is marked or unmarked within discourse
Large deviations for non-uniformly expanding maps
We obtain large deviation results for non-uniformly expanding maps with
non-flat singularities or criticalities and for partially hyperbolic
non-uniformly expanding attracting sets. That is, given a continuous function
we consider its space average with respect to a physical measure and compare
this with the time averages along orbits of the map, showing that the Lebesgue
measure of the set of points whose time averages stay away from the space
average decays to zero exponentially fast with the number of iterates involved.
As easy by-products we deduce escape rates from subsets of the basins of
physical measures for these types of maps. The rates of decay are naturally
related to the metric entropy and pressure function of the system with respect
to a family of equilibrium states. The corrections added to the published
version of this text appear in bold; see last section for a list of changesComment: 36 pages, 1 figure. After many PhD students and colleagues having
pointed several errors in the statements and proofs, this is a correction to
published article answering those comments. List of main changes in a new
last sectio
The compound Poisson limit ruling periodic extreme behaviour of non-uniformly hyperbolic dynamics
We prove that the distributional limit of the normalised number of returns to
small neighbourhoods of periodic points of non-uniformly hyperbolic dynamical
systems is compound Poisson. The returns to small balls around a fixed point in
the phase space correspond to the occurrence of rare events, or exceedances of
high thresholds, so that there is a connection between the laws of Return Times
Statistics and Extreme Value Laws. The fact that the fixed point in the phase
space is a repelling periodic point implies that there is a tendency for the
exceedances to appear in clusters whose average sizes is given by the Extremal
Index, which depends on the expansion of the system at the periodic point.
We recall that for generic points, the exceedances, in the limit, are
singular and occur at Poisson times. However, around periodic points, the
picture is different: the respective point processes of exceedances converge to
a compound Poisson process, so instead of single exceedances, we have entire
clusters of exceedances occurring at Poisson times with a geometric
distribution ruling its multiplicity.
The systems to which our results apply include: general piecewise expanding
maps of the interval (Rychlik maps), maps with indifferent fixed points
(Manneville-Pomeau maps) and Benedicks-Carleson quadratic maps.Comment: To appear in Communications in Mathematical Physic
Mathematical Aspects of Vacuum Energy on Quantum Graphs
We use quantum graphs as a model to study various mathematical aspects of the
vacuum energy, such as convergence of periodic path expansions, consistency
among different methods (trace formulae versus method of images) and the
possible connection with the underlying classical dynamics.
We derive an expansion for the vacuum energy in terms of periodic paths on
the graph and prove its convergence and smooth dependence on the bond lengths
of the graph. For an important special case of graphs with equal bond lengths,
we derive a simpler explicit formula.
The main results are derived using the trace formula. We also discuss an
alternative approach using the method of images and prove that the results are
consistent. This may have important consequences for other systems, since the
method of images, unlike the trace formula, includes a sum over special
``bounce paths''. We succeed in showing that in our model bounce paths do not
contribute to the vacuum energy. Finally, we discuss the proposed possible link
between the magnitude of the vacuum energy and the type (chaotic vs.
integrable) of the underlying classical dynamics. Within a random matrix model
we calculate the variance of the vacuum energy over several ensembles and find
evidence that the level repulsion leads to suppression of the vacuum energy.Comment: Fixed several typos, explain the use of random matrices in Section
On the Lebesgue measure of Li-Yorke pairs for interval maps
We investigate the prevalence of Li-Yorke pairs for and
multimodal maps with non-flat critical points. We show that every
measurable scrambled set has zero Lebesgue measure and that all strongly
wandering sets have zero Lebesgue measure, as does the set of pairs of
asymptotic (but not asymptotically periodic) points.
If is topologically mixing and has no Cantor attractor, then typical
(w.r.t. two-dimensional Lebesgue measure) pairs are Li-Yorke; if additionally
admits an absolutely continuous invariant probability measure (acip), then
typical pairs have a dense orbit for . These results make use of
so-called nice neighborhoods of the critical set of general multimodal maps,
and hence uniformly expanding Markov induced maps, the existence of either is
proved in this paper as well.
For the setting where has a Cantor attractor, we present a trichotomy
explaining when the set of Li-Yorke pairs and distal pairs have positive
two-dimensional Lebesgue measure.Comment: 41 pages, 3 figure
The acquisition of Sign Language: The impact of phonetic complexity on phonology
Research into the effect of phonetic complexity on phonological acquisition has a long history in spoken languages. This paper considers the effect of phonetics on phonological development in a signed language. We report on an experiment in which nonword-repetition methodology was adapted so as to examine in a systematic way how phonetic complexity in two phonological parameters of signed languages â handshape and movement â affects the perception and articulation of signs. Ninety-one Deaf children aged 3â11 acquiring British Sign Language (BSL) and 46 hearing nonsigners aged 6â11 repeated a set of 40 nonsense signs. For Deaf children, repetition accuracy improved with age, correlated with wider BSL abilities, and was lowest for signs that were phonetically complex. Repetition accuracy was correlated with fine motor skills for the youngest children. Despite their lower repetition accuracy, the hearing group were similarly affected by phonetic complexity, suggesting that common visual and motoric factors are at play when processing linguistic information in the visuo-gestural modality
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